The number sequence game starts with a sequence $S$ of $N$ numbers written on a line.

Two players alternate turns. The players on their respective turns must select and remove either the first or the last number remaining in the sequence.

A player's own score is (determined by) the sum of all the numbers that player has taken. Each player attempts to maximize their own sum.

If $N=4$ and $S=\{1,2,10,3\}$, then each player maximizes their own score as follows:

• Player 1: removes the first number ($1$)
• Player 2: removes the last number from the remaining sequence ($3$)
• Player 1: removes the last number from the remaining sequence ($10$)
• Player 2: removes the remaining number ($2$)

Player 1 score is $1+10=11$.

Let $F(N)$ be the score of player 1 if both players follow the optimal strategy for the sequence $S=\{s_1,s_2,\dots ,s_N\}$ defined as:

• $s_1=0$
• $s_{i+1}=(s_i^2+45)$ modulo $1000000007$

The sequence begins with $S=\{0,45,2070,4284945,753524550,478107844,894218625,\dots \}$.

You are given $F(2)=45$, $F(4)=4284990$, $F(100)=26365463243$, $F({10}^4)=2495838522951$.

Find $F({10}^8)$.